Yes there are seasonal allergies in Israel too. Israeli trees also make pollen, and since the blossoming season starts sometime in shevat, and winter frost that usually puts an end to summer allergies doesn’t really ever come, seasonal allergies sort of become all-year round allergies. and then there are all those sandstorms, not great at all for those with respiratory problems. B”H for allergy medications!
It can take time to develop an allergy – so that someone moving to a new place/climate might be fine in the beginning, but after several years, the allergic reaction starts again.